93 lines
4.6 KiB
TeX
93 lines
4.6 KiB
TeX
\documentclass[../schrodinger_simulation.tex]{subfiles}
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\begin{document}
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\section{Results}\label{sec:results}
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\subsection{Deviation}\label{ssec:deviation}
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% Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
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We used the \verb|superlu| solver, which is a solver for sparse matrices. It is
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generally used to solve nonsymmetric, sparse matrices. However, as the alternative
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solver \verb|lapack| converts a sparse matrix to a dense matrix, it will increase
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memory usage compared to \verb|superlu|.
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% Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
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% Add plot of deviation for both single- and double-slit
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Since we used a solver for sparse matrices, we decrease the number of computations performed
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compared to number of computations using a solver for dense matrices.
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We checked if the total probability was conserved over time, by plotting the deviation
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from $1.0$.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/probability_deviation.pdf}
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\caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.}
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\label{fig:deviation}
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\end{figure}
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In Figure \ref{fig:deviation}, we observe a larger deviation of total probability
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for a barrier with double slits compared to no barrier. Interaction with the
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barrier result in a change ... of values, as the light passes through,
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The result is more prone to computational errors. Using no barrier, the light is not
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affected by obstacles resulting in a more stable deviation from the total probability.
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In addition, we have to consider the limitation of a computer, some computational
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error is to be expected.
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\subsection{Time evolution}\label{ssec:time_evolution}
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% Problem 8: Colormap, include plot of both Re and Im for different time steps
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% Account for color scale
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We ran the simulation using the values in column 2 of Table \ref{tab:sim_setup}, found
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in Section \ref{ssec:implementation}. To study the time evolution of the probability
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function, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob},
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Figure \ref{fig:colormap_1_prob}, and Figure \ref{fig:colormap_2_prob} show the
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result for time steps $t=[0, 0.001, 0.002]$, respectively. In addition, we created
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separate plots for the real and imaginary part of $u_{\ivec, \jvec}$, for the same
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time steps. The result can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_0_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0$.}
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\label{fig:colormap_0_prob}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_1_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.001$.}
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\label{fig:colormap_1_prob}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_2_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.}
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\label{fig:colormap_2_prob}
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\end{figure}
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In Figure \ref{fig:colormap_1_prob}, the ... interacts with the double
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slitted barrier, which result in a wavelike pattern. However, the waves are more
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visible when we observe the real and imaginary part separately in Figure \ref{fig:colormap}.
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\subsection{Particle detection}\label{ssec:particle_detection}
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% Problem 9: Plot detection probability for single-, double- and triple-slit
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We used the simulation from the previous section, assumed a detector screen
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located at $x=0.8$, and plotted the detection probability along the screen at time
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$t=0.002$. We adjusted the parameters to include single-, double-, and triple-slitted
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barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{fig:}.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/single_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a single-slit barrier.}
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\label{fig:particle_detection_single}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/double_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a double-slit barrier.}
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\label{fig:particle_detection_double}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/triple_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a triple-slit barrier.}
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\label{fig:particle_detection_triple}
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\end{figure}
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\end{document}
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